Hypothesis Test: Difference Between Proportions

This lesson explains how to conduct a hypothesis test to determine
whether the difference between two proportions is significant.
The test procedure, called the
two-proportion z-test, is
appropriate when the following conditions are met:

Each sample includes at least 10 successes and 10 failures.
(Some texts say that 5 successes and 5 failures are enough.)

Each population is at least 10 times as big as its sample.

This approach consists of four steps: (1) state the hypotheses,
(2) formulate an analysis plan, (3) analyze sample data, and
(4) interpret results.

State the Hypotheses

Every hypothesis test requires the analyst
to state a
null hypothesis
and an
alternative hypothesis.
The table below shows three sets of hypotheses. Each makes a
statement about the difference d between
two population proportions, P1 and P2.
(In the table, the symbol ≠ means " not equal to ".)

Set

Null hypothesis

Alternative hypothesis

Number of tails

1

P1 - P2 = 0

P1 - P2 ≠ 0

2

2

P1 - P2> 0

P1 - P2 < 0

1

3

P1 - P2< 0

P1 - P2 > 0

1

The first set of hypotheses (Set 1) is an example of a
two-tailed test, since an extreme value on either side of the
sampling distribution would cause a researcher to reject the null
hypothesis. The other two sets of hypotheses (Sets 2 and 3) are
one-tailed tests, since an extreme value on only one side of the
sampling distribution would cause a researcher to reject the
null hypothesis.

When the null hypothesis states that there is no difference between the
two population proportions (i.e., d = 0), the null and alternative
hypothesis for a two-tailed test are often stated in the
following form.

H0: P1 = P2
Ha: P1 ≠ P2

Formulate an Analysis Plan

The analysis plan describes
how to use sample data to accept or reject the null
hypothesis. It should specify the following elements.

Significance level. Often, researchers choose
significance levels
equal to
0.01, 0.05, or 0.10; but any value between 0 and
1 can be used.

Test method. Use the two-proportion z-test (described in the
next section) to determine whether the hypothesized
difference between population proportions differs
significantly from the observed sample difference.

Analyze Sample Data

Using sample data, complete the following computations to find
the test statistic and its associated P-Value.

Pooled sample proportion. Since the null hypothesis states
that P1=P2, we use a pooled sample
proportion (p) to compute the
standard error
of the sampling distribution.

p = (p1 * n1 + p2 * n2)
/ (n1 + n2)

where p1 is the sample proportion from population 1,
p2 is the sample proportion from population 2,
n1 is the size of sample 1, and
n2 is the size of sample 2.

Standard error. Compute the standard error (SE)
of the sampling distribution difference between two proportions.

SE =
sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

where p is the pooled sample proportion,
n1 is the size of sample 1, and
n2 is the size of sample 2.

Test statistic. The test statistic is a z-score (z) defined by
the following equation.

z = (p1 - p2) / SE

where p1 is the proportion from sample 1,
p2 is the proportion from sample 2, and
SE is the standard error of the sampling distribution.

P-value. The P-value is the probability of observing a
sample statistic as extreme as the test statistic. Since the
test statistic is a z-score, use the
Normal Distribution Calculator
to assess the probability associated with the z-score. (See
sample problems at the end of this lesson for examples of how this
is done.)

The analysis described above is a two-proportion z-test.

Interpret Results

If the sample findings are unlikely, given
the null hypothesis, the researcher rejects the null hypothesis.
Typically, this involves comparing the P-value to the
significance level,
and rejecting the null hypothesis when the P-value is less than
the significance level.

Test Your Understanding

In this section, two sample problems illustrate how to conduct a
hypothesis test for the difference between two proportions.
The first problem involves a a
two-tailed test; the second problem, a one-tailed test.

Problem 1: Two-Tailed Test

Suppose the Acme Drug Company develops a new drug, designed to
prevent colds. The company states that the drug is equally effective
for men and women. To test this claim, they choose a
a simple random sample of 100 women and 200 men from a population of
100,000 volunteers.

At the end of the study, 38% of the women caught a cold; and
51% of the men caught a cold. Based on these findings,
can we reject the company's claim that the drug is equally
effective for men and women? Use a 0.05 level of significance.

Solution: The solution to this problem takes four steps:
(1) state the hypotheses, (2) formulate an analysis plan,
(3) analyze sample data, and (4) interpret results.
We work through those steps below:

State the hypotheses. The first step is to
state the null hypothesis and an alternative hypothesis.

Null hypothesis: P1 = P2
Alternative hypothesis: P1 ≠ P2

Note that these hypotheses constitute a two-tailed test.
The null hypothesis will be rejected if the proportion
from population 1
is too big or if it is too small.

Formulate an analysis plan. For this analysis,
the significance level is 0.05. The test method is a
two-proportion z-test.

Analyze sample data. Using sample data, we
calculate the pooled sample proportion (p) and the standard error
(SE). Using those measures, we compute the z-score
test statistic (z).

Interpret results. Since the P-value (0.034) is
less than the significance level (0.05), we cannot accept the
null hypothesis.

Note: If you use this approach on an exam, you may also want to mention
why this approach is appropriate.
Specifically, the approach is appropriate because
the sampling method was simple random sampling, the
samples were independent, each population was at least 10 times
larger than its sample, and
each sample included at least 10 successes and 10 failures.

Problem 2: One-Tailed Test

Suppose the previous example is stated a little bit differently.
Suppose the Acme Drug Company develops a new drug, designed to
prevent colds. The company states that the drug is more effective
for women than for men. To test this claim, they choose a
a simple random sample of 100 women and 200 men from a population of
100,000 volunteers.

At the end of the study, 38% of the women caught a cold; and
51% of the men caught a cold. Based on these findings,
can we conclude that the drug is more effective for women than
for men? Use a 0.01 level of significance.

Solution: The solution to this problem takes four steps:
(1) state the hypotheses, (2) formulate an analysis plan,
(3) analyze sample data, and (4) interpret results.
We work through those steps below:

State the hypotheses. The first step is to
state the null hypothesis and an alternative hypothesis.

Null hypothesis: P1 >= P2
Alternative hypothesis: P1 < P2

Note that these hypotheses constitute a one-tailed test.
The null hypothesis will be rejected if the proportion
of women catching cold (p1) is sufficiently
smaller than the proportion of men catching cold
(p2).

Formulate an analysis plan. For this analysis,
the significance level is 0.01. The test method is a
two-proportion z-test.

Analyze sample data. Using sample data, we
calculate the pooled sample proportion (p) and the standard error
(SE). Using those measures, we compute the z-score
test statistic (z).

Interpret results. Since the P-value (0.017) is
greater than the significance level (0.01), we cannot reject the
null hypothesis.

Note: If you use this approach on an exam, you may also want to mention
why this approach is appropriate.
Specifically, the approach is appropriate because
the sampling method was simple random sampling, the
samples were independent, each population was at least 10 times
larger than its sample, and
each sample included at least 10 successes and 10 failures.